Appendix B: A Brief Review of the Time Value of Money Concepts

Basic Cash Flow Patterns A single cash payment/receipt at some specified point in time; a lump sum. Ordinary annuity: A series of equal cash payments/receipts occurring at the end of equal intervals of time. Annuity due: A series of equal cash payments/receipts occurring at the beginning of equal intervals of time. Three basic cash flow patterns are associated with long-term assets and liabilities:

Time Value of Money Problems: The Variables n = number of annuity payment periods i = the effective interest rate per period Pmt, payment = annuity payment per period FVn, future value = one lump sum amount (at the ending point of an n-pay annuity) that is equivalent to the annuity’s series of payments. PVn, present value = one lump sum amount (at the starting point of an n-pay annuity) that is equivalent to the annuity’s series of payments.

Mathematics of Finance: Wilson’s Octothorpe of The 4 Cases! n,i FVPV FVnPVn LUMP SUM SERIES OF PMTS BACK IN TIME FWD IN TIME NEEDED!

Notes to Students on Capital Budgeting Problems The PVn is also referred to as “the discounted future cash flow.” In theory, it should be the current “bid price” (the FMV) of a productive asset, to an individual, specific buyer. Each bidder used his/her own demanded discount rate since buyers seldom expect or demand identical rates of return on investments. Be sure to read the instruction manual on your specific brand of financial calculator carefully to make sure that you know how to input amounts.

Types of Problems The two most common time-value-of-money problems in intermediate accounting are: –to compute a present value, PVn. –to compute the interest rate, i, being earned per period. Note: If an annuity due is involved, notations will so Indicate, e.g., PVn(due) or FVn(due), as a general rule.

The Two Basic Scenerios Given a stream of estimated future cash flows and an effective interest rate, the PVn computation removes (“squeezes out”) the interest component of the business deal involved. Given a stream of estimated future cash flows and a PVn, there is “some” interest rate that makes the two equivalent to each other. In other words, an interest rate or return on investment (ROI) does exist! (Sometimes, it’s negative!)

Compute Present Value, given Pmt, demanded i and FVn Values Ajjax Co. is offering a “10% bond” investment It pays $100,000 at the end of each 6 months for 10 years, and $2,000,000 at the end of 10 years. (The “annual stated rate,” by definition, is 5%.) Assume the financial marketplaces offer similar bonds that have a 12% stated rate. Likely, Pat, an investor, therefore, will demand an ROI of 6% semiannually from Ajjax on this particular bond. Can you compute the present value (to Pat) of the bond’s expected cash receipts?

The PVn of a given deal = the PVn of the stream of income checks while holding the investment PLUS the PV of the lump sum at maturity (sale) of the deal. n = 10 x 2 = 20 semiannual periods. i = 6%, which is the given rate per period. Pmt (in an ordinary annuity) = $100,000 FV (the lump sum future value) = $2,000,000 Can you calculate the PVn of the deal = ? (The FMV today is $1,770,602)

Compute the Rate of Interest PVn of the deal, price paid = $1,770,602 n = 10 x 2 = 20 semiannual periods. Pmt (in an ordinary annuity) = $100,000 FV (the lump sum future value) = $2,000,000 Can you calculate the ROI of the deal = ? (i = 6% each six months!) Note: The annual effective rate (APR) is more than 12%. What is it, and why?

Compute Present Value An auto lease on a Ford Explorer calls for 36 monthly payments of $800 per month (with the first payment due upon signing and all future payments due on the first day of each month); there is no other down payment. There is a $5,000 purchase option at the end of the lease. Assume that the lessee intends to purchase the auto at the end of the lease term; the market rate of interest on the lease is 1% per month. What is the present value of the lease payments, including the anticipated option purchase price?

The present value of $27,821 is computed as follows: n = 36 monthly payment periods i = 1%, which is given in the problem Pmt, payment (annuity due) = $800 FV (a lump sum) = $5,000 PVn(due) of the deal = ? ($27,281) –Note: Read the instruction manuals for your financial calculators to make sure you know how to set up an annuity due problem

Computing [Annual] Effective Rates of Interest A mother loans her daughter $1,500,000; the daughter signs a 3-year note payable with the following terms: $2,000,000 due at maturity (with no interest rate specified and no payments to be made during the 3 year period). What is the annual rate of interest being paid? Unless otherwise specified, if payments are not scheduled in interim time frames of less than a year, annual periods are assumed; and, each year follows the I = Prt formula.

The effective rate of interest = 10.06% per year, based on … n = 3 periods Periodic payment = $0 Maturity value = $2,000,000 Present value = $1,500,000 i for each year [period] = ? (10.06% per year)